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How to Calculate Frequency Response Manually

A child on a swing, a radio tuner, a skyscraper in an earthquake ̵2; they are all examples of systems responding to a frequency. Although the details of each are different, the mathematics that describe their response to an input force are all the same. When the force is in the form of an oscillating input, the response will depend upon the difference between the frequency of the input force and the natural frequency of the system. Even when the force isn̵7;t strictly periodic, the response can still be represented in terms of the sum of the response to the different frequencies that make up the input force. That̵7;s why understanding the frequency response is so important.

Instructions

- 1
  Measure the natural motion of your system. If your system is a bell, you̵7;d give it a tap and measure the loudness and pitch of the sound; if it̵7;s a pendulum you̵7;d swing it back and let it go and measure the time it takes to swing and how large an angle it swings through. For example, you could pull a baseball attached to a spring down from its rest position and find that it returns to the bottom every 1 1/4 seconds and that the maximum distance from its rest position decreases by 1/2 after 20 seconds.
- 2
  Calculate the resonant frequency of your system. This is the frequency at which it will run if it̵7;s displaced once and left to move on its own. For the example system, the time it takes to complete a bounce is 1.25 seconds, so the resonant frequency is given by 1/1.25 seconds = 0.8 per second. It will be convenient to label this f0.
- 3
  Calculate the damping constant of the system. The damping constant measures how much the system ̶0;winds down̶1; after it̵7;s given a little bump. It̵7;s given by the equation:
  damping = - (2 / (t1 - t0)) x ln (amplitude(t1)/amplitude(t0)); where t1 and t2 are the measurement times, and the amplitudes are measured at their maximum. For the example, the initial measurement was at time 0 and the final measurement at time = 20 sec and the amplitude ratio was 0.5, so the damping is:
  damping = - (2/20) x ln(0.5) = .069 per second.
- 4
  Identify the magnitude and frequency of the forcing function. The forcing function can be a radio transmission, the wind blowing over a bridge or a child rotating the end of a jump rope. For the example, assume your spring is attached to a plate on the ceiling, and you move the plate up and down with a frequency of 0.5 per second through a distance of 5 cm. The full displacement distance is twice the amplitude, so the magnitude of the forcing function is 2.5 cm.
- 5
  Calculate the response of the system to the forcing function. The response is given by:
  response(time) = A0 x cos(ff x time - phase) where A0 is the magnitude of the motion, ff is the frequency of the forcing function, and phase represents the time delay of the response. A0 and the phase are given by:
  A0 = f0^2 x force amplitude/sqrt((f0^2 - ff^2)^2 + damping^2 x ff^2)
  phase= arctan (damping x ff/(f0^2 - ff^2)).
  For the example,
  A0 = 0.8^2 x 2.5/sqrt ((0.8^2 - 0.5^2)^2 + .069^2 x 0.5^2) = 4.1 cm
  phase = arctan (0.069 x 0.5/(0.8^2 - 0.5^2)) = 0.09;
  So the response of the system to a frequency force is
  response(time) = 4.1 cm x cos (0.5 x time - 0.09).

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